Question

In a sample of 388 people in a certain, 22% had red hair. Find a 99% confidence interval for the proportion of people in the country that have red hair.

Answer 1

Answer: (0.1658, 0.2742)

Step-by-step explanation:

Formula to find the confidence interval for population proportion is given by :-

$\hat{p}\pm z^*\sqrrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$

, where  n= sample size

z*= Critical value

$\hat{p}$ = sample proportion.

As per given , we have

Significance level : $\alpha=1-0.99=0.01$

According to z-table, Critical value for 99% confidence interval : z*=2.576

Let p be the proportion of people in the country that have red hair.

n= 388

$\hat{p}=0.22$

Now, required confidence interval for proportion of people in the country that have red hair will be :-

$0.22\pm (2.576)\sqrt{\dfrac{0.22(1-0.22)}{388}}$

$0.22\pm (2.576)\sqrt{0.000442268}$

$0.22\pm (2.576)(0.02103)$

$(0.22-0.05417328,\ 0.22+0.05417328 )=(0.16582672,\ 0.27417328)\\\\\approx(0.1658,\ 0.2742)$

The 99% confidence interval for the proportion of people in the country that have red hair= (0.1658, 0.2742)